Marcus Scheiblecker

Chain-Linking in Quarterly National Accounts and the Business Cycle

The 2008 World Congress on National Accounts and Economic Performance Measures for Nations

Washington D. C., 12th to 17th May 2008May 14 Session 3 E: Expert Seminar Session on Index Number

TheoryKey Bridge Marriott, The Francis Scott Key Ballroom Salon A

19.04.23

Whereas chain-linking of annuals in previous years prices is unambiguous, it is not at quarterly frequencies. Contrary to the US, annuals as well as quarters are calculated at average prices of the whole previous year in the EU. For the consecutive construction of quantity indexes, there exist three different methods:

The over-the-year method

OTYsy

syy

syy

OTYsy I

qp

qpI ,1

,11

,1

,

19.04.23

The annual-overlap method

The quarterly-overlap method

AOsy

syy

syy

AOsy I

qp

qpI ,1

,11

,1

,

4

QOy

yy

syy

QOsy I

qp

qpI 4,1

4,11

,1

,

19.04.23

Three different chain-linking methods

Austrian GDP in million €

Date nominal (at prices of the current quarter)

at average prices of the current year

at average prices of the previous year

2004-01 55316.88 55883.83 54749.82

2004-02 58206.69 58323.31 57223.87

2004-03 60165.13 60055.64 59066.67

2004-04 62129.84 61555.77 60727.75

2005-01 57257.14 57433.55 56310.32

2005-02 60614.11 60743.55 59604.40

2005-03 62513.27 62528.67 61398.34

2005-04 64718.30 64397.05 63324.27

2006-01 59452.31   58996.74

2006-02 63595.43   62623.76

2006-03 65441.57   64489.25

2006-04 67899.57   66494.52

blue = over-the-year method green = quarterly-overlap method red = annual-overlap method

19.04.23

Time series properties

Over-the-year method: Breaks in the series (compared to the previous quarter) occur every quarter. Does not represent a time series in a narrower statistical sense.

Annual-overlap method: Breaks in the series (compared to the previous quarter) occur every first quarter of a year. Represents a time series in a narrower statistical sense only within a year.

Quarterly-overlap method: No breaks in the series (compared to the previous quarter) occur. Represents a time series in a narrower statistical sense.

19.04.23

Time consistency (additivity) property

Over-the-year method: approximately time consistent even away from the reference period

Annual-overlap method: fully time consistent

Quarterly-overlap method: not time consistent, especially away from the reference period

⇨ Splitting-up annual discrepancies over quarters by a method generating time series in a narrower sense (proportional Denton procedure, spline functions) does not interfere with the time series properties of the benchmarked series.

Note: Quarterly-overlap method + pro-rata distribution of annual discrepancies ≙ annual-overlap method. According to the IMF‘s Quarterly National Accounts Manual (2001, p. 84): ‘Because of the step problem, the pro-rata distribution technique is not acceptable.‘

19.04.23

Differences between the sum of the quarters and annual data in Austrian

GDP

M illion € Percent M illion € Percent € mn Percent

1988 12.02 0.01 0.00 0.00 – 128.41 – 0.09

1989 12.94 0.01 0.00 0.00 – 229.65 – 0.15

1990 11.94 0.01 0.00 0.00 422.48 0.26

1991 12.43 0.01 0.00 0.00 761.13 0.45

1992 4.99 0.00 0.00 0.00 730.39 0.42

1993 10.23 0.01 0.00 0.00 839.55 0.48

1994 4.91 0.00 0.00 0.00 497.05 0.28

1995 7.97 0.00 0.00 0.00 266.50 0.15

1996 7.60 0.00 0.00 0.00 98.26 0.05

1997 9.34 0.00 0.00 0.00 – 266.37 – 0.14

1998 11.24 0.01 0.00 0.00 – 721.54 – 0.37

1999 9.50 0.00 0.00 0.00 – 1,587.38 – 0.78

2000 0.00 0.00 0.00 0.00 ± 0.00 0.00

2001 – 19.63 – 0.01 0.00 0.00 591.10 0.28

2002 – 20.52 – 0.01 0.00 0.00 1,006.43 0.47

2003 – 22.55 – 0.01 0.00 0.00 1,458.10 0.67

2004 5.57 0.00 0.00 0.00 1,457.55 0.66

2005 19.59 0.01 0.00 0.00 636.15 0.28

2006 23.59 0.01 0.00 0.00 286.39 0.12

OTY AO QO

19.04.23

Relative differences between the AO- and the Denton benchmarked QO-

method

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06

Percent

19.04.23

Distribution of annual chain-linking-differences by the AO and the B-QO-

method

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06

AO B-QO

Percent

19.04.23

Consequences for time series modelling of QNA data chain-linked by

different methods

Why time series modelling of QNA series?

Outlier detection procedures for preparing time series for a following seasonal adjustment are based on time series analysis.

For seasonal adjustment an extrapolation of the series beyond the time series horizon is necessary to apply filter techniques for the recent observations (which are in the focus of interest). All extrapolation methods rely on time series properties.

For TRAMO-SEATS the seasonal component is extracted by factorization of the time series model. For X-12 instead, mathematical filters are applied (making this procedure slightly less dependent on neat time series properties).

For some kind of business cycle analysis (Beveridge-Nelson decomposition, unobserved components models, …)

19.04.23

Models suggested by TRAMO-SEATS

Series Span (n° of obs.) Q1.1988 - Q4.2006 (76)Model Span (n° of obs.) Q1.1988 - Q3.2006 (75)Method Tramo/SeatsPRE-ADJUSTMENTTransformation LogarithmMean Correction NoneCorrection for Trading Day Effects 1 Regressor(s) (holiday-corr.) Trad1 t-value 1.06 [-1.990, 1.990] 5% Trad2 t-value -1.06 (derived) [-1.990, 1.990] 5%Correction for Easter Effect NoneCorrection for Outliers Autom.:AO,LS,TC Critical t-value 3.063Corr. for Missing Obs. NoneCorr. for Other Regr. Effects NoneSpecif. of the ARIMA model (0 1 1)(0 1 1) (fixed) Non-seas. MA (lag 1) value -0.3500 Non-seas. MA (lag 1) t-value -3.05 [-1.990, 1.990] 5% Seasonal MA (lag 4) value -0.2378 Seasonal MA (lag 4) t-value -1.91 [-1.990, 1.990] 5%Method of Estimation Exact Maximum LikelihoodDECOMPOSITIONARIMA Decomposition ExactSeasonality Seasonal model used

Information on Models Model 1 (Tramo-Seats)Over-the-year method

19.04.23

Series Span (n° of obs.) Q1.1988 - Q4.2006 (76)Model Span (n° of obs.) Q1.1988 - Q4.2006 (76)Method Tramo/SeatsPRE-ADJUSTMENTTransformation LogarithmMean Correction NoneCorrection for Trading Day Effects NoneCorrection for Easter Effect NoneCorrection for Outliers Autom.:AO,LS,TC; 1 Outlier(s) fixed Critical t-value 3.065 AO Q1.1993 t-value -3.88 [-3.065, 3.065] crit.val.Corr. for Missing Obs. NoneCorr. for Other Regr. Effects NoneSpecif. of the ARIMA model (0 1 1)(0 1 1) (fixed) Non-seas. MA (lag 1) value -0.2551 Non-seas. MA (lag 1) t-value -2.18 [-1.990, 1.990] 5% Seasonal MA (lag 4) value -0.2533 Seasonal MA (lag 4) t-value -2.16 [-1.990, 1.990] 5%Method of Estimation Exact Maximum LikelihoodDECOMPOSITIONARIMA Decomposition ExactSeasonality Seasonal model used

Information on Models Model 1 (Tramo-Seats)Annual-overlap method

19.04.23

Series Span (n° of obs.) Q1.1988 - Q4.2006 (76)Model Span (n° of obs.) Q1.1988 - Q4.2006 (76)Method Tramo/SeatsPRE-ADJUSTMENTTransformation LogarithmMean Correction NoneCorrection for Trading Day Effects 1 Regressor(s) (holiday-corr.) Trad1 t-value 0.99 [-1.990, 1.990] 5% Trad2 t-value -0.99 (derived) [-1.990, 1.990] 5%Correction for Easter Effect NoneCorrection for Outliers Autom.:AO,LS,TC; 1 Outlier(s) fixed Critical t-value 3.065 TC Q1.1993 t-value -4.09 [-3.065, 3.065] crit.val.Corr. for Missing Obs. NoneCorr. for Other Regr. Effects NoneSpecif. of the ARIMA model (0 1 1)(0 1 1) (fixed) Non-seas. MA (lag 1) value -0.2567 Non-seas. MA (lag 1) t-value -2.22 [-1.990, 1.990] 5% Seasonal MA (lag 4) value -0.2242 Seasonal MA (lag 4) t-value -1.83 [-1.990, 1.990] 5%Method of Estimation Exact Maximum LikelihoodDECOMPOSITIONARIMA Decomposition ExactSeasonality Seasonal model used

Information on Models Model 1 (Tramo-Seats)Quarterly-overlap method

19.04.23

Series Span (n° of obs.) Q1.1988 - Q4.2006 (76)Model Span (n° of obs.) Q1.1988 - Q4.2006 (76)Method Tramo/SeatsPRE-ADJUSTMENTTransformation LogarithmMean Correction NoneCorrection for Trading Day Effects NoneCorrection for Easter Effect NoneCorrection for Outliers Autom.:AO,LS,TC; 1 Outlier(s) fixed Critical t-value 3.065 AO Q1.1993 t-value -4.22 [-3.065, 3.065] crit.val.Corr. for Missing Obs. NoneCorr. for Other Regr. Effects NoneSpecif. of the ARIMA model (0 1 0)(0 1 1) (fixed) Seasonal MA (lag 4) value -0.2720 Seasonal MA (lag 4) t-value -2.38 [-1.990, 1.990] 5%Method of Estimation Exact Maximum LikelihoodDECOMPOSITIONARIMA Decomposition ExactSeasonality Seasonal model used

Information on Models B-QO (Tramo-Seats)Denton benchmarked quarterly-overlap method

19.04.23

Overview over time series models

time-series model

outlier detected

OTY (0,1,1)(0,1,1) noneAO (0,1,1)(0,1,1) AO 1993Q1QO (0,1,1)(0,1,1) TC 1993Q1B-QO (0,1,0)(0,1,1) AO 1993Q1

19.04.23

Quarter-to-quarter percentage changes

seasonally and working day adjusted

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

1999 2000 2001 2002 2003 2004 2005 2006

OTY AO B-QO

19.04.23

BC extraction

Seasonally adjusted OTY, AO and B-QO series(by time-series modelling techniques)HP1600-filtered

Unadjusted series (not modelled)BK 32-8 filterd

19.04.23

Austrian chained GDP HP-1600 filtered

GDPqo : HPX Transformed SeriesGDPao : HPX Transformed SeriesGDPoty : HPX Transformed Series

1-1988 4-1989 3-1991 2-1993 1-1995 4-1996 3-1998 2-2000 1-2002 4-2003 3-2005

1.1001.000

900800700600500400300200100

0-100-200-300-400-500-600-700

19.04.23

Log differencies ofHP-1600-transformed series

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

1999 2000 2001 2002 2003 2004 2005 2006

OTY AO B-QO

19.04.23

Co-movement of cyclical components

Series CoherenceAverage Spectru

m

Mean Delay

Cross-correlation

2 Y-8 Y 2 Y-8 Y 2 Y-8 Y r0 rmax tmax (1)

GDPao 0.98 0.41 -0.01 0.96 0.96 0

GDPqo 0.88 0.42 0.1 0.92 0.92 0

Note: The + (-) sign refers to a lead (lag) vis-à-vis the reference series.

19.04.23

Bry-Boschan BC turning pointsin HP-1600 filtered series

Peak Trough Peak Trough Peak Trough Peak Trough

Reference series 1991Q3 1993Q1 1994Q1 1995Q1 1996Q1 1997Q2 2000Q2 2003Q4

AO 0 – – 0 0 – 1 0 0 – 1

B-QO 0 – – 0 0 – 1 0 0 – 1

OTY 0 – – 0 0 0 0 – 1 – 1

Note: The + (-) sign refers to a lag (lead) vis-à -vis the reference series.

Extra cyc les

19.04.23

Business cycle turning points

1992 1993 1994 1995 1996 1997

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Austria

HP   ▲           ▼         ▲                 ▼    

BK ▲           ▼           ▲                 ▼    

HP Dyn Fact ▲       ▼             ▲                        

BK Dyn Fact             ▼         ▲                        

Breuss (1984)                                                

Hahn et (1992)                                                

Artis et M. (2004)                   ▼       ▲             ▼      

Artis et K. (2004)             ▼                                  

Euro area

HP               ▼         ▲             ▼        

BK             ▼             ▲             ▼      

Mönch et (2004) ▲       ▼                                      

Forni et (2000)                 ▼       ▲         ▼            

Artis et M. (2004)               ▼         ▲             ▼        

Artis et K. (2004)   ▲       ▼                                    

CEPR (2003)       ▼               ▲       ▼               ▲

19.04.23

Capacity utilisation in the Austrian industry

75

80

85

90

1987 1989 1991 1993 1995 1997 1999 2001 2003

Percent

19.04.23

Baxter-King 32-8-filtered series

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

1999 2000 2001 2002 2003 2004 2005 2006

OTY AO B-QO

19.04.23

Bry-Boschan BC turning points for BK 32-8 filtered series

Peak Trough Peak Trough Peak Trough Peak Trough Peak Trough

1988Q4 1990Q1 1991Q4 1993Q2 1996Q1 1997Q2 2000Q1 2003Q2 2004Q3 2006Q1

AO 0 0 0 0 0 0 2 0 0 0 0

B-QO 0 0 0 0 0 0 2 0 0 0 0

OTY 0 0 0 0 0 0 2 0 0 – 2 0

Note: The + (-) sign refers to a lag (lead) vis-à -vis the reference series.

Extra cyc les

Reference series

19.04.23

Conclusions

Different quarterly chain-linking methods generate different time series.

Their different time series properties can potentially interfere with modelling outliers, seasonalfactors,BC-components, …

This can lead to different results for analysis based on model pre-processed series.

The turning point detection process itself (not basedon data pre-processed or pre-adjusted by timeseries models) seems to be rather robust to differentquarterly chain-linking methods.